The spaces on the left and right of
have equivalent norms, the intersection of sets corresponds to summation of
norms. They also coincide as sets. So see this it suffices to use the
proposition (
Tensor product of
function spaces
) to construct bases for dense subsets.

Proposition

(Stability of splitting for Sobolev spaces with dominating
mixed derivative) Assume the condition
(
Sparse tensor product setup
).
Then for
,
,
we
have
for a
decomposition